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Theorem un0.1 39775
Description: is the constant true, a tautology (see df-tru 1657). Kleene's "empty conjunction" is logically equivalent to . In a virtual deduction we shall interpret to be the empty wff or the empty collection of virtual hypotheses. in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
un0.1.1 (      ▶   𝜑   )
un0.1.2 (   𝜓   ▶   𝜒   )
un0.1.3 (   (      ,   𝜓   )   ▶   𝜃   )
Assertion
Ref Expression
un0.1 (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1
StepHypRef Expression
1 un0.1.1 . . . 4 (      ▶   𝜑   )
21in1 39557 . . 3 (⊤ → 𝜑)
3 un0.1.2 . . . 4 (   𝜓   ▶   𝜒   )
43in1 39557 . . 3 (𝜓𝜒)
5 un0.1.3 . . . 4 (   (      ,   𝜓   )   ▶   𝜃   )
65dfvd2ani 39569 . . 3 ((⊤ ∧ 𝜓) → 𝜃)
72, 4, 6uun0.1 39774 . 2 (𝜓𝜃)
87dfvd1ir 39559 1 (   𝜓   ▶   𝜃   )
Colors of variables: wff setvar class
Syntax hints:  wtru 1654  (   wvd1 39555  (   wvhc2 39566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-an 386  df-tru 1657  df-vd1 39556  df-vhc2 39567
This theorem is referenced by:  sspwimpVD  39915
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